Problem with partially UV-unwrapping a simple object
I found the problem.

Apparrently in 2.8 you can't just add image texture nodes and link everything up like I used to do in older Blender versions. Blender will only read a texture if it's imported through the properties panel, next to the color picker button. Thought I'd try to do the exact same steps that are shown in the tutorial (even though they result in exactly the same node structure) and it worked, but trying to build an identical node setup somehow fails...